Computation of Moore-Penrose generalized inverses of matrices with meromorphic function entries

J.R. Sendra is member of the Research Group ASYNACS (Ref.CT-CE2019/683)In this paper, given a field with an involutory automorphism, we introduce the notion of Moore-Penrose field by requiring that all matrices over the field have Moore-Penrose inverse. We prove that only characteristic zero fields can be Moore-Penrose, and that the field of rational functions over a Moore-Penrose field is also Moore-Penrose. In addition, for a matrix with rational functions entries with coefficients in a field K, we find sufficient conditions for the elements in K to ensure that the specialization of the Moore-Penrose inverse is the Moore-Penrose inverse of the specialization of the matrix. As a consequence, we provide a symbolic algorithm that, given a matrix whose entries are rational expression over C of finitely many meromeorphic functions being invariant by the involutory automorphism, computes its Moore-Penrose inverve by replacing the functions by new variables, and hence reducing the problem to the case of matrices with complex rational function entries.Ministerio de Economía y CompetitividadEuropean Regional Development Fun

Cissoid constructions of augmented rational ruled surfaces

J.R. Sendra is member of the Research Group ASYNACS (Ref.CT-CE2019/683)Given two real affine rational surfaces we derive a criterion for deciding the rationality of their cissoid. Furthermore, when one of the surfaces is augmented ruled and the other is either an augmented ruled or an augmented Steiner surface, we prove that the cissoid is rational. Furthermore, given rational parametrizations of the surfaces, we provide a rational parametrization of the cissoid.Ministerio de Economía y CompetitividadEuropean Regional Development Fun

Distance bounds of ϵ-points on hypersurfaces

ϵ-points were introduced by the authors (see [S. Pérez-Díaz, J.R. Sendra, J. Sendra, Parametrization of approximate algebraic curves by lines, Theoret. Comput. Sci. 315(2–3) (2004) 627–650 (Special issue); S. Pérez-Díaz, J.R. Sendra, J. Sendra, Parametrization of approximate algebraic surfaces by lines, Comput. Aided Geom. Design 22(2) (2005) 147–181; S. Pérez-Díaz, J.R. Sendra, J. Sendra, Distance properties of ϵ-points on algebraic curves, in: Series Mathematics and Visualization, Computational Methods for Algebraic Spline Surfaces, Springer, Berlin, 2005, pp. 45–61]) as a generalization of the notion of approximate root of a univariate polynomial. The notion of ϵ-point of an algebraic hypersurface is quite intuitive. It essentially consists in a point such that when substituted in the implicit equation of the hypersurface gives values of small module. Intuition says that an ϵ-point of a hypersurface is a point close to it. In this paper, we formally analyze this assertion giving bounds of the distance of the ϵ-point to the hypersurface. For this purpose, we introduce the notions of height, depth and weight of an ϵ-point. The height and the depth control when the distance bounds are valid, while the weight is involved in the bounds.Ministerio de Educación y CienciaComunidad de MadridUniversidad de Alcal

Parametrization of aproximate algebraic surfaces by lines

In this paper we present an algorithm for parametrizing approximate algebraic surfaces by lines. The algorithm is applicable to ²-irreducible algebraic surfaces of degree d having an ²–singularity of multiplicity d−1, and therefore it generalizes the existing approximate parametrization algorithms. In particular, given a tolerance ² > 0 and an ²-irreducible algebraic surface V of degree d, the algorithm computes a new algebraic surface V , that is rational, as well as a rational parametrization of V . In addition, in the error analysis we show that the output surface V and the input surface V are close. More precisely, we prove that V lies in the offset region of V at distance, at most, O(² 1 2d )

On the approximate parametrization problem of algebraic curves.

The problem of parameterizing approximately algebraic curves and surfaces is an active research field, with many implications in practical applications. The problem can be treated locally or globally. We formally state the problem, in its global version for the case of algebraic curves (planar or spatial), and we report on some algorithms approaching it, as well as on the associated error distance analysis

An algorithm to parametrize approximately space curves

This is the author’s\ud version of a work that was accepted for publication in\ud Journal of Symbolic Computation. Changes resulting from the publishing\ud process, such as peer review, editing, corrections,\ud structural formatting, and other quality control mechanisms may not be\ud reflected in this document.\ud Changes may have been made to this work since it was submitted for\ud publication.\ud A definitive version was subsequently published in Journal of Symbolic\ud Computation vol. 56 pp. 80-106 (2013).\ud DOI: 10.1016/j.jsc.2013.04.002We present an algorithm that, given a non-rational irreducible\ud real space curve, satisfying certain conditions, computes a rational\ud parametrization of a space curve near the input one. For a given\ud tolerance \epsilon > 0, the algorithm checks whether a planar projection\ud of the given space curve is \epsilon -rational and, in the affirmative\ud case, generates a planar parametrization that is lifted to a space\ud parametrization. This output rational space curve is of the same\ud degree as the input curve, both have the same structure at infinity,\ud and the Hausdorff distance between their real parts is finite.\ud Moreover, in the examples we check that the distance is small.This work has been developed, and partially supported, by the Spanish “Ministerio de Ciencia e\ud Innovación” under the Project MTM2008-04699-C03-01, and by the “Ministerio de Economía y Competitividad”\ud under the project MTM2011-25816-C02-01. All authors belong to the Research Group\ud ASYNACS (Ref. CCEE2011/R34)

Radical parametrization of algebraic curves and surfaces

Parametrization of algebraic curves and surfaces is a fundamental topic in CAGD (intersections; offsets and conchoids; etc.) There are many results on rational parametrization, in particular in the curve case, but the class of such objects is relatively small. If we allow root extraction, the class of parametrizable objetcs is greatly enlarged (for example, elliptic curves can be parametrized with one square root). We will describe the basics and the state of the art of the problem of parametrization of curves and surfaces by radicals.Junta de Extremadura and FEDER fundsThis contribution is partially supported by the Ministerio de Econom´ıa y Competitividad under the project MTM2011-25816-C02-01, by the Austrian Science Fund (FWF) P22766-N18, and by Junta de Extremadura and FEDER funds. The first author is a member of the of the Research Group asynacs (Ref. ccee2011/r34)

Bounding and Estimating the Hausdorff distance between real space algebraic curves

This is the author’s version of a work that was accepted for publication in Computational and Applied Mathematics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Rueda S.L., Sendra J., Sendra J.R., (2014). "Bounding and Estimating the Hausdorff distance\ud between real space algebraic curves ". Computer Aided Geometric Design. vol 31 (2014)\ud 182-198; DOI 10.1016/j.cagd.2014.02.005In this paper, given two real space algebraic curves, not necessarily bounded,\ud whose Hausdor distance is nite, we provide bounds of their distance. These\ud bounds are related to the distance between the projections of the space curves onto\ud a plane (say, z = 0), and the distance between the z-coordinates of points in the\ud original curves. Using these bounds we provide an estimation method for a bound\ud of the Hausdor distance between two such curves and we check in applications that\ud the method is accurate and fas

Number of common roots and resultant of two tropical univariate polynomials

It is well known that for two univariate polynomials over the complex number field the number of their common roots is equal to the order of their resultant. In this paper, we show that this fundamental relationship still holds for the tropical polynomials under suitable adaptation of the notion of order, if the roots are simple and non-zero.Ministerio de Economía y CompetitividadEuropean Regional Development FundH. Hong is partially supported by US NSF 131963

Radical parametrization of algebraic curves and surfaces

The first author is a member of the Research Group asynacs (Ref. ccee2011/r34).Parametrization of algebraic curves and surfaces is a fundamental topic in CAGD (intersections; offsets and conchoids; etc.) There are many results on rational parametrization, in particular in the curve case, but the class of such objects is relatively small. If we allow root extraction, the class of parametrizable objetcs is greatly enlarged (for example, elliptic curves can be parametrized with one square root). We will describe the basics and the state of the art of the problem of parametrization of curves and surfaces by radicals.Ministerio de Economía y CompetitividadAustrian Science Fund (FWF